Key to "Proved the trigonometry identity csc2xcot2+sec2x+1=cos2x"

Tabansi

Answered question

2021-09-03

Proved the trigonometry identity

\frac{\csc^{2} x}{\cot^{2} + \sec^{2} x + 1} = \cos^{2} x

hajavaF

Skilled2021-09-04Added 90 answers

Given:

\frac{\csc^{2} x}{\cot^{2} + \sec^{2} x + 1} = \cos^{2} x

To prove:
The given identity.
Consider, left hand side of the given,

\frac{\csc^{2} x}{\cot^{2} + \sec^{2} x + 1}

Using the trigonometric identity:

\cot^{2} x + 1 = \csc^{2} x

\Rightarrow \frac{\csc^{2} x}{\cot^{2} x + \sec^{2} x + 1} = \frac{\csc^{2} x}{\csc^{2} x + \sec^{2} x}

Using:

\csc^{2} x = \frac{1}{\sin^{2} x}

\sec^{2} x = \frac{1}{\cos^{2} x}

\Rightarrow \frac{\csc^{2} x}{\cot^{2} x + \sec^{2} x + 1} = \frac{(\frac{1}{\sin^{2} x})}{(\frac{1}{\sin^{2} x} + \frac{1}{\cos^{2} x})}

= \frac{(\frac{1}{\sin^{2} x})}{(\frac{\cos^{2} x + \sin^{2} x}{\sin^{2} x \cos^{2} x})}

= \frac{1}{\sin^{2} x} \cdot \frac{\sin^{2} x \cos^{2} x}{(\cos^{2} x + \sin^{2} x)}

= \frac{\cos^{2} x}{\cos^{2} x + \sin^{2} x}

Using the identity:

\cos^{2} x + \sin^{2} x = 1

\Rightarrow \frac{\csc^{2} x}{\cot^{2} x + \sec^{2} x + 1} = \frac{\cos^{2} x}{1} = 1

Hence the proof.

Jeffrey Jordon

Expert2022-01-31Added 2605 answers

Answer is given below (on video)

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